A characterization of Gaussian measures on locally compact Abelian groups

Let $\xi$ and $\eta$ be independent random variables having equal variance. In order that $\xi+\eta$ and $\xi-\eta$ be independent, it is necessary and sufficient that $\xi$ and $\eta$ have normal distributions. This result of Bernshtein [1] is carried over in [7] to the case when $\xi$ and $\eta$ take values in a locally compact Abelian group. In the present note, a characterization of Gaussian measures on locally compact Abelian groups is given in which in place of $\xi+\eta$ and $\xi-\eta$, functions of $\xi$ and $\eta$ are considered which satisfy the associativity equation.